Patterns of neural activity underlie human cognition. Transitions between these patterns are orchestrated by the brain's network architecture. What are the mechanisms linking network structure to cognitively relevant activation patterns? Here we implement principles of network control to investigate how the architecture of the human connectome shapes transitions between 123 experimentally defined cognitive activation maps (cognitive topographies) from the NeuroSynth meta-analytic engine. We also systematically incorporate neurotransmitter receptor density maps (18 receptors and transporters) and disease-related cortical abnormality maps (11 neurodegenerative, psychiatric and neurodevelopmental diseases; N = 17 000 patients, N = 22 000 controls). Integrating large-scale multimodal neuroimaging data from functional MRI, diffusion tractography, cortical morphometry, and positron emission tomography, we simulate how anatomically-guided transitions between cognitive states can be reshaped by pharmacological or pathological perturbation. Our results provide a comprehensive look-up table charting how brain network organisation and chemoarchitecture interact to manifest different cognitive topographies. This computational framework establishes a principled foundation for systematically identifying novel ways to promote selective transitions between desired cognitive topographies.